Borsuk–Ulam theorem
Theorem in topology
Borsuk–Ulam theorem
In topology, the Borsuk–Ulam theorem states that every continuous map from the sphere to the plane maps some pair of antipodal points to the same point. The theorem is named after mathematicians Karol Borsuk and Stanisław Ulam. Informally, for a "balloon animal" (or any arbitrarily distorted shape) made out of a spherical balloon, if it is squashed into a plane (letting the air out somehow), then at least one pair of points that were opposite each other on the original sphere will be squashed onto the same point of the plane. An analogous statement is true in higher dimensions (see below).
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